Suppose , are finite-dimensional vector spaces over a field, with dimensions and , respectively. For any space let denote the space of linear operators on . The partial trace over , Tr, is a mapping
Let and be bases for and respectively. Then has a matrix representation where and relative to the basis of the space given by . Consider the sum
for over . This gives the matrix . The associated linear operator on is independent of the choice of bases and is defined as the partial trace.
Consider a quantum system, , in the presence of an environment, . Consider what is known in quantum information theory as the CNOT gate:
Suppose our system has the simple state and the environment has the simple state . Then . In the quantum operation formalism we have
where we inserted the normalization factor .